WAVES OF MAGNETIC FLUX 41 



two oppositely rotating waves of half the amplitude is a reciprocal 

 one ; i.e. any rotating wave may be analyzed into two simple alter- 

 nating waves of half the amplitude. The proof of this proposition is 

 \tremely simple. For, taking the rotating wave 



u = U cos (pt qx) 

 we have 



u = U cos pt cos qx + U sin pt sin qx 



But each of the terms on the right-hand side represents an alter- 

 nating wave, and the maximum amplitude of each alternating wave 

 is half the amplitude of the given rotating wave. Thus the pro- 

 position is established. 



The substitution of two rotating waves for an alternating one, or 

 rice versa, is a device frequently employed in alternating-current 

 theory, as it in many cases leads to a considerable simplification in 

 the treatment of certain classes of problems. 



20. Production of Rotating Waves of Magnetic 

 Flux by Means of Polyphase Currents 



We shall now show that a rotating wave of magnetic flux may be 

 produced by using suitably arranged windings supplied with poly- 

 phase currents. 



Let us suppose that a two-phase system of mains is available. 

 Imagine a winding embedded in the stator (or rotor) which, when 

 traversed by a simple alternating current obtained from one phase 

 of the two-phase system of mains generates a simple alternating 

 wave of magnetic flux represented by 



y\ = B sin pt sin qx 



2-jr 2tr 



where, as before, p = ^ and q = -j-, T and X denoting the period 



and the wave-length respectively of the wave of magnetic flux. 



Let another winding, precisely similar to the first, but displaced 

 relatively to it by A (or half a pole-pitch), be arranged on the stator, 

 and let a current be sent through it from the remaining phase of the 

 two-phase system. This winding will generate a simple alternating 

 magnetic flux wave precisely similar to that generated by the first 

 winding, but displaced relatively to it as regards both time and space. 



For sin pt we must now, by reason of the phase difference of - c 



& 



which exists between the two currents, write sin \pt + j, and 



